Maximum Flow is Fair: A Network Flow Approach to Committee Voting

Abstract: In the committee voting setting, a subset of $k$ alternatives is selected based on the preferences of voters. In this paper, our goal is to efficiently compute $\textit{ex-ante}$ fair probability distributions over committees. We introduce a new axiom called $\textit{group resource proportionality}$, which strengthens other fairness notions in the literature. We characterize our fairness axiom by a correspondence with max flows on a network formulation of committee voting. Using the connection to flow networks revealed by this characterization, we introduce two voting rules which achieve fairness in conjunction with other desiderata. The first rule - the $\textit{redistributive utilitarian rule}$ - satisfies ex-ante efficiency in addition to our fairness axiom. The second rule - Generalized CUT - reduces instances of our problem to instances of the minimum-cost maximum flow problem. We show that Generalized CUT maximizes social welfare subject to our fairness axiom and additionally satisfies an incentive compatibility property known as $\textit{excludable strategyproofness}$. Lastly, we show our fairness property can be obtained in tandem with strong $\textit{ex-post}$ fairness properties - an approach known as $\textit{best-of-both-worlds}$ fairness. We strengthen existing best-or-both-worlds fairness results in committee voting and resolve an open question posed by Aziz et al. [2023].